In a known grinding apparatus of this type (DE-OS No. 29 12 814) the holding and guiding part is a cylindrical spindle sleeve, surrounding the grinding sleeve in the area of a center hole of the workpiece. The grinding spindle carries on one of its ends the grinding tool, adapted to the inclination of a valve seat surface. This known grinding apparatus no longer satisfies the high accuracy requirements for workpieces with a relatively small center hole, the diameter of which is approximately 6 mm. or less, for example. The reason for this is the lack of stability of the arrangement. There is an increasing need to produce workpieces with very small center bores, for example, for fuel injection pump nozzle bodies.